Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Modern Physics Letters B

The renormalization of geometric quantum discord in the transverse Ising model

Xiu-Xing Zhang, Hongrong Li

Published Jan 30, 2015• 4 citations
Publisher

Abstract

Using the quantum renormalization group (QRG) method, we investigate the ground state geometric measure of quantum discord (GQD) of the transverse Ising model. The analytical expression of GQD together with the RG recurrence relations are obtained. The critical and scaling behaviors of GQD are investigated. Our results show that with the increasing of RG steps, GQD (first derivative of GQD) exhibits sudden change (dip) in behavior at the critical point, which indicates that the system undergoes phase transitions between the Ising and the paramagnetic phases. The critical exponent that governs the divergences of the first derivative of GQD is calculated. How the critical point of the model is approached as the increasing of the system size is revealed.

DOI
10.1142/s0217984915500025

Related papers

Quantum Renormalization Group for Ground State Quantum Discord

Javad Vahedi

Using the quantum renormalization group, we have studied a quantum phase transition via studying the ground state quantum discord. Our results show that the characterization of quantum criticality, as well as the scaling analysis of relevant properties with system size for the Ising model in the transverse field based on the quantum renormalization group, have a similar behavior to the concurrence, which was studied in Ref. [Phys. Rev. A 76, 060304(R) (2007)].

Chinese Journal of Physics•Jun 1, 2015•0 citations

Quantum Renormalization of Spin Squeezing in Spin Chains

Leila Balazadeh, G. Najarbashi, Ali Tavana

By employing quantum renormalization group (QRG) method, we investigate quantum phase transitions (QPT) in the Ising transverse field (ITF) model and in the XXZ Heisenberg model, with and without Dzyaloshinskii Moriya (DM) interaction, on a periodic chain of N lattice sites. We adopt a new approach called spin squeezing as an indicator of QPT. Spin squeezing, through analytical expression of a spin squeezing parameter, is calculated after each step of QRG. As the scale of the system becomes larger, (after many QRG steps), the ground state (GS) spin squeezing parameters show an abrupt change at a quantum critical point (QCP). Moreover, in all of the studied models, the first derivative of the spin squeezing parameter with respect to the control parameter is discontinuous, which is a signature of QPT. The spin squeezing parameters develop their saturated values after enough QRG iterations. The divergence exponent of the first derivative of the spin squeezing parameter in the near vicinity of the QCP is associated with the critical exponent of the correlation length.

Scientific Reports•Dec 6, 2018•10 citations

Entanglement and quantum phase transition in the Heisenberg—Ising model

Xiaodong Tan, Bai-Qi Jin, Wei Gao

We use the quantum renormalization-group (QRG) method to study the entanglement and quantum phase transition (QPT) in the one-dimensional spin-1/2 Heisenberg—Ising model [Lieb E, Schultz T and Mattis D 1961 Ann. Phys. (N.Y.) 16 407]. We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations. We also investigate the scaling behavior of the system close to the quantum critical point, which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size. Also, the first derivative of concurrence between two blocks diverges at the quantum critical point, which is directly associated with the divergence of the correlation length.

Chinese Physics B•Feb 1, 2013•7 citations

NEGATIVITY AND GEOMETRIC QUANTUM DISCORD AS INDICATORS OF QUANTUM PHASE TRANSITION IN THE XY MODEL WITH DZYALOSHINSKII–MORIYA INTERACTION

Shuai Xu, XUE KE SONG, Liu Ye

In this paper, we use the negativity and geometric quantum discord (GQD) to investigate the quantum phase transition (QPT) in the anisotropic spin-1/2 XY model with staggered Dzyaloshinskii–Moriya (DM) interaction using the quantum renormalization-group method, and the results show that the negativity and GQD can both obtain the quantum critical points associated with QPTs after several iterations of the renormalization. In addition to this, we discuss how the strength of the DM interaction and anisotropic parameter make the effect on the negativity and GQD for different RG steps. At last, we give out the relationship between negativity and GQD, and verify that with anisotropy parameter and the strength of DM interaction increased, the squared negativity and the normalization of GQD tend to equal in this model.

International Journal of Modern Physics B•May 20, 2013•7 citations

Thermodynamic properties of transverse field quantum Ising model using tensor network formalism / Pang Sin Yang

S. Y. Pang

Ising model has been successful in describing ferromagnetism and its phase transition to paramagnet. At the critical point, the free energy density function and its derivatives diverge. Their behaviour near the critical point are described by power-laws with associated critical exponents. In many cases, the critical exponents can be determined from analytic solutions via conventional renormalization groups methods or from Monte Carlo simulations. However, for quantum many-body systems, very few are tractable to analytical solutions. The quantum many-body wavefunction belongs to large dimensional Hilbert space that increases exponentially with system size. If the Hamiltonian is gapped and only local interaction is considered, then the wavefunction can be efficiently truncated. Tensor Network formalism provides a scheme to truncate the less important degrees of freedom via Singular Value Decomposition (SVD) of the density matrix. In this study, we investigated the thermodynamic properties and phase transition of one-dimensional transverse-field quantum Ising model (1D-tQIM) under the finite-size effect and random coupling strength. Starting with Matrix Product States (MPS) as a wavefunction ansatz, the Density Matrix Renormalization Group algorithm is applied to the MPS. The variational algorithm, which iteratively performs SVDs and truncation at each bond, approximates the ground state MPS wavefunction. All quantum observables are calculated from the contraction of the resultant ground state MPS. Although theoretically, divergence at critical points only happen in an infinite system, one can obtain the critical exponents through simulation of finite-size 1D-tQIM. Using the analytic solution as a benchmark, we compared the finite-size effects of the system using finite-size scaling analysis and MPS methods. The critical exponents of 1D-tQIM are independently calculated and compared with the analytical results. Thermodynamic quantities such as magnetization, susceptibility and correlation function are calculated for system sizes of 20, 40, 60, 80, 100 and 120 spins. We determined the respective critical exponents:

•Mar 1, 2020•0 citations

Strong-disorder renormalization group study of aperiodic quantum Ising chains

Fleury J Oliveira Filho, Maicon S Faria, André P. Vieira

We employ an adaptation of a strong-disorder renormalization group technique in order to analyze the ferro–paramagnetic quantum phase transition of Ising chains with aperiodic but deterministic couplings under the action of a transverse field. In the presence of marginal or relevant geometric fluctuations induced by aperiodicity, for which the critical behavior is expected to depart from the Onsager universality class, we derive analytical and asymptotically exact expressions for various critical exponents (including the correlation length and the magnetization exponents, which are not easily obtainable by other methods), and shed light onto the nature of the ground-state structures in the neighborhood of the critical point. The main results obtained by this approach are confirmed by finite-size scaling analyses of numerical calculations based on the free-fermion method.

arXiv:1112.5141•Journal of Statistical Mechanics Theory and Experiment•Mar 12, 2012•12 citations
Using the quantum renormalization group (QRG) method, we investigate the ground state geometric measure of quantum discord (GQD) of the transverse Ising model. The analytical expression of GQD together with the RG recurrence relations are obtained. The critical and scaling behaviors of GQD are investigated. Our results show that with the increasing of RG steps, GQD (first derivative of GQD) exhibits sudden change (dip) in behavior at the critical point, which indicates that the system undergoes phase transitions between the Ising and the paramagnetic phases. The critical exponent that governs the divergences of the first derivative of GQD is calculated. How the critical point of the model is approached as the increasing of the system size is revealed.
By employing quantum renormalization group (QRG) method, we investigate quantum phase transitions (QPT) in the Ising transverse field (ITF) model and in the XXZ Heisenberg model, with and without Dzyaloshinskii Moriya (DM) interaction, on a periodic chain of N lattice sites. We adopt a new approach called spin squeezing as an indicator of QPT. Spin squeezing, through analytical expression of a spin squeezing parameter, is calculated after each step of QRG. As the scale of the system becomes larger, (after many QRG steps), the ground state (GS) spin squeezing parameters show an abrupt change at a quantum critical point (QCP). Moreover, in all of the studied models, the first derivative of the spin squeezing parameter with respect to the control parameter is discontinuous, which is a signature of QPT. The spin squeezing parameters develop their saturated values after enough QRG iterations. The divergence exponent of the first derivative of the spin squeezing parameter in the near vicinity of the QCP is associated with the critical exponent of the correlation length.
We use the quantum renormalization-group (QRG) method to study the entanglement and quantum phase transition (QPT) in the one-dimensional spin-1/2 Heisenberg—Ising model [Lieb E, Schultz T and Mattis D 1961 Ann. Phys. (N.Y.) 16 407]. We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations. We also investigate the scaling behavior of the system close to the quantum critical point, which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size. Also, the first derivative of concurrence between two blocks diverges at the quantum critical point, which is directly associated with the divergence of the correlation length.
In this paper, we use the negativity and geometric quantum discord (GQD) to investigate the quantum phase transition (QPT) in the anisotropic spin-1/2 XY model with staggered Dzyaloshinskii–Moriya (DM) interaction using the quantum renormalization-group method, and the results show that the negativity and GQD can both obtain the quantum critical points associated with QPTs after several iterations of the renormalization. In addition to this, we discuss how the strength of the DM interaction and anisotropic parameter make the effect on the negativity and GQD for different RG steps. At last, we give out the relationship between negativity and GQD, and verify that with anisotropy parameter and the strength of DM interaction increased, the squared negativity and the normalization of GQD tend to equal in this model.
Ising model has been successful in describing ferromagnetism and its phase transition to paramagnet. At the critical point, the free energy density function and its derivatives diverge. Their behaviour near the critical point are described by power-laws with associated critical exponents. In many cases, the critical exponents can be determined from analytic solutions via conventional renormalization groups methods or from Monte Carlo simulations. However, for quantum many-body systems, very few are tractable to analytical solutions. The quantum many-body wavefunction belongs to large dimensional Hilbert space that increases exponentially with system size. If the Hamiltonian is gapped and only local interaction is considered, then the wavefunction can be efficiently truncated. Tensor Network formalism provides a scheme to truncate the less important degrees of freedom via Singular Value Decomposition (SVD) of the density matrix. In this study, we investigated the thermodynamic properties and phase transition of one-dimensional transverse-field quantum Ising model (1D-tQIM) under the finite-size effect and random coupling strength. Starting with Matrix Product States (MPS) as a wavefunction ansatz, the Density Matrix Renormalization Group algorithm is applied to the MPS. The variational algorithm, which iteratively performs SVDs and truncation at each bond, approximates the ground state MPS wavefunction. All quantum observables are calculated from the contraction of the resultant ground state MPS. Although theoretically, divergence at critical points only happen in an infinite system, one can obtain the critical exponents through simulation of finite-size 1D-tQIM. Using the analytic solution as a benchmark, we compared the finite-size effects of the system using finite-size scaling analysis and MPS methods. The critical exponents of 1D-tQIM are independently calculated and compared with the analytical results. Thermodynamic quantities such as magnetization, susceptibility and correlation function are calculated for system sizes of 20, 40, 60, 80, 100 and 120 spins. We determined the respective critical exponents:
We employ an adaptation of a strong-disorder renormalization group technique in order to analyze the ferro–paramagnetic quantum phase transition of Ising chains with aperiodic but deterministic couplings under the action of a transverse field. In the presence of marginal or relevant geometric fluctuations induced by aperiodicity, for which the critical behavior is expected to depart from the Onsager universality class, we derive analytical and asymptotically exact expressions for various critical exponents (including the correlation length and the magnetization exponents, which are not easily obtainable by other methods), and shed light onto the nature of the ground-state structures in the neighborhood of the critical point. The main results obtained by this approach are confirmed by finite-size scaling analyses of numerical calculations based on the free-fermion method.